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Let $A$ be an $n$ by $n$ matrix and $x$ a free parameter. Define $$p(A,x)=\sum_{\pi \in S_n} x^{n(\pi)}A_{1\pi(1)}\ldots A_{n\pi(n)},$$ where $\pi$ ranges over the permutation group $S_n$ and $n(\pi)$ is the number of cycles in the cycle decomposition of $\pi$. Clearly, $p(A,1)=perm(A)$, the permanent. In general, $p(A,x)$ has properties in common with the permanent such as $p(PAQ,x)=p(A,x)$ for permutation matrices $P,Q$.

Is this a well-known structure in combinatorics and where might I find more information?

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3 Answers 3

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The collection of immanants

$$Imm_\lambda(A) = \sum_{\pi \in S_m} \chi_\lambda(\pi) A_{i\pi(1)}...A_{n\pi(n)}$$

generalized the permanent and determinant and your $p(A,x)$, since we can write $x^{n(\pi)}$ as a linear combination of characters.

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    $\begingroup$ Continuing Doug's answer, when we write $x^{n(\pi)}$ as a linear combination of irreducible characters $\chi^\lambda$, then the coefficient of $\chi^\lambda$ is $\prod_u (x+c(u))/h(u)$, where $u$ runs over all squares in the diagram of $\lambda$, $c(u)$ is the content of $u$, and $h(u)$ is the hook length of $u$. $\endgroup$
    – Richard Stanley
    Commented Mar 15, 2010 at 18:29
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This exact version of a "generalized permanent" is called $\beta$-extension in this Foata-Zeilberger paper (see also my paper for the algebraic context and further non-commutative generalizations, Cartier-Foata style).

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There are some papers on this topic from the point of view of rook theory by V. S. Shevelev.

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